Long-Period Wave Responses in a Harbor with Narrow Mouth
Weon Mu Jeong1, Kil Seong Lee2 and Woo Sun Park1
Abstract
Field measurements of long-period wave have been preformed for the
Gamcheon Harbor in Korea. A Galerkin finite element model based on the
extended mild-slope equation has been developed for simulating harbor oscillation
more accurately. Infinite and joint elements are introduced to accomodate
radiation condition at infinity and matching conditions at harbor mouth for the
consideration of the energy loss due to flow separation, respectively. Comparisons
with the results obtained by hydraulic experiments by Lepelletier (1980) show that
the present model gives fairly good results. Model results reveal that the influence
of entrance loss at the harbor mouth is considerably significant. From the
application to the Gamcheon Harbor it is seen that the computed resonant
periods and amplification ratios well agree with the measured results. The
entrance loss effects were found to be insignificant unless the incident wave
height is large.
Introduction
Long-period harbor oscillations could create unacceptable vessel movements
leading to the downtime of moored ships.
It is practically very difficult to
prevent long-period harbor oscillations, but extension of breakwaters at the harbor
mouth could be a countermeasure in part. Narrowing a harbor mouth might give
rise to increase in the energy loss due to flow separation near the mouth, which
in turn makes resonant periods of the harbor become longer, especially for the
Helmholtz resonant mode. Referring to Lepelletier (1980), the reduction of a
harbor mouth can reduce the amplification ratios of Helmholtz mode considerably.
'Senior Research Scientist, Coastal and Harbor Engineering Research Center, Korea Ocean Research
and Development Institute, 1270 Sa-dong, Ansan, Kyeonggi-do, 425-170, Korea
Professor, School of Civil, Urban and Geo-Systems Engineering, Seoul National University,
San56-1, Sinrim-dong, Kwanak-ku, Seoul, 151-742, Korea
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COASTAL ENGINEERING 1998
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This study includes field measurements for short- and long-period waves
around the Gamcheon Harbor in Korea, and the development of a finite element
model based on the extended mild-slope equation, which incorporates the bottom
frictional dissipation and the entrance loss due to flow separation. The present
model can also handle the harbor resonance problems in a harbor with
non-straight coastlines. The model is verified through the comparisons with
Lepelletier (1980)'s experimental results. Finally, the model is applied to the
Gamcheon Harbor with narrow mouth and compared with field measurements.
Field Measurement
Field measurements for short- and long-period waves have been performed
around Gamcheon Harbor which is located at the east-southern coast of Korea.
The harbor is 4,000 m long, 1,150 m wide and has entrance of 240 m wide. In
spite of narrow mouth, this commercial harbor is suffering from severe downtime
problems in summer season. The measurement period was from November 27 to
December 13 in 1997. Four pressure-type wave gauges (PWG), and one Aanderaa
RCM-9 current meter were deployed at locations shown in Figure 1. Sampling
intervals for the pressures at Sts. PI and P4 were set to 5 seconds, while
pressure data at Sts. P2 and P3 were gathered in the interval of 1 second.
Current velocities from RCM-9 were averaged over 1 minute.
Figure 1. Location Map of Field Measurements.
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After filtering tidal components from pressure signals using a Butterworth
high-pass filter in MATLAB, spectral analyses of the 16 sets of filtered pressure
data were performed. Figure 3 shows the power spectral densities of long-period
wave data set no. 6 obtained at the four stations. We can find the first and
second resonant modes appear near 2,000 seconds and 600—700 seconds,
respectively.
It is noted that the period of Helmholtz resonant mode of
Gamcheon Harbor is 27.0~33.3 minutes, and the second and third resonant
periods appear at 9.4 — 12.1 and 5.2~6.2 minutes, respectively. Figure 3 shows
the power spectral densities of current velocity normal to the harbor mouth. It is
noted that the maximum value appears around 28.2 — 31.9 minutes corresponding
to the Helmholz mode.
1E+1 -s: No. 6 (Dec. 2 13:00, 1997)
St. P1
"" 1E+0
St. P2
£• 1E-1
1E-4
i
i i i i n
-i—i—i—rr
100
1000
Wave period (s)
Figure 2. Power Spectral Densities Obtained at Sts. PI—P4.
1000
Wave period (s)
Figure 3. Power Spetral Densities for Observed Velocity at St. C.
COASTAL ENGINEERING 1998
1185
Galerkin Finite Element Model
Theoretical Formulation
A Cartesian coordinate system (x, y) and a cylindrical coordinate system
( r, 6) are employed for mathematical formulation. The domain in the model is
divided into two regions as shown in Figure 4. One is a near field region (i2x)
that is modeled as conventional finite elements and the other is a far field region
(i32) that is represented as infinite elements of which shape functions satisfy the
radiation condition at infinity. In the far field region, the water depth is assumed
to be constant in the radial direction, but the depth in the circumferential direction
varies with the value of the interface between the far and near field regions, JT/.
To take into account the entrance loss at the narrow harbor mouth, the near field
region is again divided into two sub-regions, that is inner (i21;) and outer regions
( Qi0) of the harbor (see Figure 4).
Incident wave
Figure 4. Definition Sketch for Boundary Value Problem.
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COASTAL ENGINEERING 1998
The monochromatic and simple time harmonic waves propagating over a
steeply sloped sea bed with variable depths in both regions may be described as
follows (Massel, 1993; Suh et al., 1997).
v • (CCgv<t>,) + ^-w2^- w2{Rl(vh)2 + R2v2k}<t>i = 0
(1)
in which v = (d/dx)i+ (d/9y)j, i and j are unit vectors in the directions of
x and y, respectively, 4>i is the two-dimensional spatial complex valued
velocity potential, the subscript i denotes the near field region for i = 1 and
the far field region for i = 2, co is the angular frequency, h(x, y) is the
water depth, Rx and R2 are coefficients for second-order bottom effects
corresponding to the squared bottom slope (v/z)2 and the bottom curvature
V2h (see Suh et al., 1997). Wave celerity, C and group velocity, Cg are
given as
C= \jj;tcmhkh
(2)
C,
w
2 V^ sinh2M/
in which k is the wave number.
To consider the effects of wave absorbing, the partial reflection boundary
condition is introduced along the solid boundaries, which was proposed by Mei
and Chen(1975) as a function of an empirical reflection coefficient, Kr normal
to the solid boundaries. This boundary condition is represented as
-^ = ah
dn
d(*2 + 4i)
dn
on A
= a{<j>2+4>^
on r2
(4a)
(4b)
in which 4>\ is the total velocity potential in the near field region, <f>2 is the
scattered wave potential in the far field region, <j)l is the incident wave
potential, n is outward normal to the solid boundary, and a is expressed as
a = ikcosd, i
+
%
in which 0l is the angle of the incident wave normal to the solid boundary.
(5)
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The matching boundary condition on .T7 can be expressed as
(6a)
01 = 02 + 0/
d<t>l
dn
=
3(02+ <t>l)
dn
^x
The scattered wave potential, (/>2 in the far field region must satisfy the
Sommerfeld radiation condition at infinity.
lim\^(^7 -Mi) = 0
(7)
The incident wave potential is given as
ikrcos(8— 9i)
,-Q\
in which a0 is the amplitude of incident wave and 6t is the attack angle of
incident wave.
Two matching conditions were introduced on the interface boundary of two
sub-regions, i.e., velocity and pressure continuity conditions at the harbor mouth
(Unluata and Mei, 1975). For the entrance loss effects, the following matching
conditions are introduced.
Uj =
u0
Pi
Po , 1 fe
,,, h 3U0
— = — + TJ- — u0\u0\ + -1- -jr
(9)
(10)
where, u is the flow velocity, p is the pressure, the subscript i denotes the
inner region and o denotes the outer region, p is the fluid density, fe is the
loss coefficient and /, is the jet length. The quadratic non-linear energy loss
term was linearized by using Lorentz transformation and equating depth averaged
power, that is,
y -f u0\u„\ = y a u0
(11)
where, the linearized loss coefficient a is given by
„ — 8 Ar t„„u u 5 + cosh2^fe
~ "9^ g u° tanhM 2Afc+sinh2Afc
a
n 0x
(12)
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COASTAL ENGINEERING 1998
where, u„ indicates the wave mean velocity.
In the above equations, fe and /,• are determined by hydraulic experiments
for various cross-sections. We used fe based on the inverse Strouhal number,
uel am suggested by Lepelletier (1980). Then the linearized matching condition
can be given by
d</>u
dn
=
(13)
dn
d<t>u =
1
(i
\<P\0
7
dn
w
~ 4>n)
(14)
g
in which ifu ar*d <fr\0 are complex velocity potentials of the inner and outer
regions, respectively. For the jet length, a simple formula suggested by Morse and
Ingard (1968) is used.
Finite Element Formulation
• Discretization of Fluid Domain
To discretize the fluid domain in the standard finite element manner, it is
necessary to describe the unknown potential, <j>r> m terms of the nodal potential
vector, {4>ei}' f°r
{N}, as follows:
an
*i = {N}Titi}
element (e), and the prescribed shape function vector,
(15)
Using Galerkin's technique, the boundary value problem can be re-formulated as
integral equations. Using following definition of the residual for each element
{Re) = - f ,.{Jv}[v • (CC^>d + -%-eo2^i-o>2{Ri(vh)2 + R2y2h}^]dal (16)
and Green's second identity, and introducing the above boundary conditions, the
system equation can be obtained for each element. Taking the residual as zero
gives following simultaneous equations:
£ {([**J + [Ker) + [KerJ){tf) + {F°r) + {n,,} + {F*fll}} = {0} (17)
COASTAL ENGINEERING 1998
in which [FCQ), [Ker), [KerJ, {Fer), {FerJ, and
system matrices given by for the near field region:
1189
{Fes) are the element
-a>2(% - {Ri(vh)2 + R2v2h}\{N}{N}T]di2e1
[/CV,] =
frCCga{NHMTdn
(18a)
(18b)
[K°rJ = Jr CCg -f^ {N} drmli + J CCg -^ {TV} dTMo
(18c)
{Fer) = {0}
(18d)
irra) = - Jrccgd{%\ +> man
as.)
and for the far field region:
[JTri] =
- o»2 (-§• - {Ri(vh)2 + /?2v2/*}){iV}{Mr] dQl
(19a)
{cCga{N){N)Tdn
(19b)
[#rj = 0
(19c)
{F%> = /^ CC,(a#,- ^) {M «71
(19d)
WrJ = - JrCc/(%;^ W ^
(19e)
• Finite, Infinite and Joint Elements
The inner region is modeled by using three-noded triangular elements, in
which the water depth h, the square of bottom slope (v/?)2, and the
curvature v2h are assumed to be constant for the convenience of numerical
calculation. In order to model efficiently the radiation condition at infinity, a
two-noded infinite element is developed. The shape function of the element is
given by
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COASTAL ENGINEERING 1998
{N} = Nr(&{NM)
{AT} = JV,(f)
for 0 <: £<: °°, -1 <i ri<. 1
for 0 <• £<, oo
(20a)
(20b)
in which {Ng(r/)} is the Lagrange shape function, and Nr(£) is the shape
function in the radial direction given by
l
NA& = -n£g-e
*~*
V £+ rr,
(21)
in which e is the artificial damping parameter( e < £), and rr, is the distance
to the infinite elements from the origin as shown in Figure 4. The artificial
damping parameter has been introduced to make the integration in Eq. (19e) in
the radial direction bounded. After the integration is completed analytically, the
value of e is taken to be zero. The shape function, Nr(£), in the radial
direction, except for the artificial damping parameter, have been derived from
the asymptotic expression for the first kind of Hankel's function in the
analytical boundary series solutions such as
#s<* J^eikr
(22)
The above shape functions satisfy the radiation condition at infinity.
• Matching the Inner and Outer Regions
Using the matching boundary conditions in Eq. (6), the total system
matrices can be assembled as
£ {([KeQ] + [Ker]){^} + {Fer} + {Fr,}) =
{0}
(23)
in which
[Kh] = [KeSl],
[**«,]
[Ker]={Ker1],
[K\X
(24)
[K\J
(25)
{F$ = { *-„}
(26)
{ FV,} = - J CCg-$-{N)dr} - ([ K\2] + [ KeaM«}
(27)
COASTAL ENGINEERING 1998
1191
and {^f} is the vector of the incident wave potential corresponding to the nodal
points.
Numerical Analyses and Discussions
Verification of the present model
To prove the validity of present model, numerical analyses have been
performed for a rectangular harbor used by Lepelletier (1980) in hydraulic
experiments. Two types of rectangular harbor have been tested. One is a fully
open harbor {alb = 1.0, a is width of a harbor entrance, b is width of harbor)
and the other is a partially open one {alb = 0.2). In Figures 6 and 7, numerical
results without and with entrance losses are presented with the experimental
results by Lepelletier (1980), respectively. As shown in figures, the calculated
amplification ratios considering energy loss effects due to flow separation coincide
very well with the experimental results. Neglecting energy loss effects obviously
over-estimates the ratio.
In general, the length of jet-flow, (, has been taken to be zero. As the /,•
increases, the resonant periods move toward longer periods and amplification ratios
at the resonant conditions increase. These phenomena have been confirmed in this
verification. As shown in Figure 7, amplification ratios for the case with /, =
0.0284 m are smaller than those for the case with /,• = 0- This is due to shifting
of the first resonant condition. The results considering the effects of the jet-flow
are better fitted to the measured results. As mentioned before, the length of the
jet-flow is estimated from the theoretical formula proposed by Morse and Ingard
(1968) which was derived for the narrow channel with a thin gate. It may be
required to develop a new formula for estimating the length of jet-flow at the
harbor mouth more accurately.
Application of the present model
To prove the applicability of the present model to the real case and to
investigate the characteristics of long-period oscillations in a harbor with narrow
mouth, numerical analysis was performed for the Gamcheon Harbor shown in
Figure 1. As previously mentioned, field measurements were carried out using
four PWGs and one RCM-9. For the direct comparison of the field measurements
and calculated results, the informations for the incident waves such as wave
heights and directions are estimated. However, it is nearly impossible to obtain
the accurate informations from the limited set of field measurements, especially
for long- period waves. Therefore, the incident wave angles were assumed to be
equal to the main direction of short-period waves previously measured. A wave
height for each wave frequency was determined from comparisons of the
estimated and measured results at two stations, PI and P2 at the outside of the
COASTAL ENGINEERING 1998
1192
harbor. The estimated incident long-period wave heights were in the range of 0.0
3-0.07 m.
The amplification ratios at the innermost station, P4 are plotted in Figure 8.
In this figure, hollow black circles indicate 13 measured data and black squares
indicate the estimated results. The resonant periods and amplification ratios
simulated by the present model are well agree with the measured results. To
investigate the effects of the entrance losses in the real situation, the numerical
analysis without considering the entrance losses was performed. The effects of the
entrance losses were however almost insignificant in the whole tested range,
except for the slight difference near the first resonant condition. This may be due
to the fact that the incident wave heights are small in the present case. The
entrance losses might be effective when the incident wave height is large as the
case of attacking of tsunami.
7
6
R = 5.2
5-
*
a
:
1
i
:•
io
:
: •
-1--B-D:
i
•
D
3
LaL 3-
•
„
•
ax
D
20n
•
[
a/b= 1.0. b/l = 0.2
ce loss
•
1
Lepelletier (19fSO)
•
.
Present( /,• = C m)
1
.
, !—1_
t
'
0.1
Incident wave height, floMo
Figure 6. Variation in Amplification Ratios with respect to Incident
Wave Heights for a Fully Open Rectangular Harbor {alb = 1.0).
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COASTAL ENGINEERING 1998
'
!
i
!
6 -
R = 6.0
5 4 3 -
2 -
IsL
2a0
1
<>
i
1
9,
!<>
i
o!
•:
0.
9 8 -
<>
-4- \-t--xr:.
"-D~-*-+-w-~
u
6 -
— -itr
i-Q.
a/b= 0.2, b/l = 0.2
without entrain e loss
4 -
LI
Lepelletier (1980)
3 -
O
Present( /, = 0 m)
•
Present( /, = 0.0284 m)
r
t —i— 1 1—t—1—1
I
'
0.1
Incident wave height, a0/h0
Figure 7. Variation in Amplification Ratios with respect to Incident
Wave Heights for a Partially Open Rectangular Harbor {alb = 0.2).
Conclusions
In this paper, the long-period wave oscillations in a harbor with narrow
mouth has been studied. Field measurements were performed for a harbor with
narrow mouth, Gamcheon Harbor in Korea. A Galerkin finite element model
based on the extended mild-slope equation was developed which can handle the
entrance losses due to flow separation. Verification of the present model was
proved through the comparison of the estimated and experimental results.
Comparisons of estimated and measured data in field shows that the present
model gives quite reasonable results.
The effects of the entrance losses are insignificant unless the incident wave
height is large. This may be true for tsunami. It was also found that strong
jet-flow can affect the resonant condition, i.e., the resonant periods are moving
toward longer period and amplification ratios are amplifying especially for the
resonant conditions, as the jet-flow is being strong.
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COASTAL ENGINEERING 1998
Wave period (s)
Figure 8. Comparison of Measured and Calculated Results for Gamcheon Harbor.
Acknowledgements
Partial support for this research was provided by Korea Ocean Research and
Development Institute through Project Nos. BSPE 97625-00-1055-2 and BSPE
97608-00-1052-2.
The authors would like to thank to their colleagues for
providing field data.
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